Inner Filter Effect Correction for Fluorescence Measurements in Microplates Using Variable Vertical Axis Focus

The inner filter effect (IFE) hinders fluorescence measurements, limiting linear dependence of fluorescence signals to low sample concentrations. Modern microplate readers allow movement of the optical element in the vertical axis, changing the relative position of the focus and thus the sample geometry. The proposed Z-position IFE correction method requires only two fluorescence measurements at different known vertical axis positions (z-positions) of the optical element for the same sample. Samples of quinine sulfate, both pure and in mixtures with potassium dichromate, showed a linear dependence of corrected fluorescence on fluorophore concentration (R2 > 0.999), up to Aex ≈ 2 and Aem ≈ 0.5. The correction extended linear fluorescence response over ≈98% of the concentration range with ≈1% deviation of the calibration slope, effectively eliminating the need for sample dilution or separate absorbance measurements to account for IFE. The companion numerical IFE correction method further eliminates the need for any geometric parameters with similar results. Both methods are available online at https://ninfe.science.


Instrumental parameters
Fluorescence and absorbance measurements were performed using the Tecan Spark M10 multimode microplate reader (Tecan, Austria). Fluorescence intensity was measured for λ ex = 345 nm and λ em = 390 nm using z-position values in the range of 14.6 -21.0 mm (Table S1). The absorbance values for both wavelengths were measured to obtain the values A ex and A em (Eq. 1), respectively in UV-transparent microplates ( Figure S11). Instrument settings of the microplate reader can be found in Table S3, SI. The solution volume in each microplate well was 200 µL. The distance from the bottom of the microplate well to the surface of the liquid, h, (Figure 1) was estimated for Greiner microplates by measuring the absorbance of pure water. The values of h for Tecan plates were measured using transparent microplates of the same geometry (transparent, 96-well, flat bottom, cat. no. 30122304, Tecan, Austria), allowing a correct calculation of k = 20.593 mm, which was used in calculations. Full details of the measurement of parameter h and specific values of geometric parameters in Figure 1 and Eq. 3, can be found in Table S2. Required geometric parameters of the microplate reader sample compartment and optical element were kindly provided by the manufacturer.

General remarks
All titrations in all experiments were performed by pipetting into 0.65 mL centrifuge tubes. Liquid handling robot Opentrons OT-2 (Opentrons, USA) was used to prepare all samples from stock solutions. Volumes of less than or equal to 30 µL were dispensed into a larger volume of solution and then 30 µL of the solution was aspirated and dispensed again to rinse the tip. After pipetting, all tubes were capped and thoroughly mixed on a vortex mixer. After mixing, aliquots of 200 µL were transferred to microplates for measurement, again using the robot. Prior to measurement, microplates were centrifuged for 2 min at 2550 rpm using a microplate centrifuge (Benchmark Scientific, USA) with additional shaking in the microplate reader sample compartment for 5 s at 1440 rpm and amplitude of 1 mm.

Solutions a) 0.05 M sulfuric acid
For all solutions containing QS and PD, 0.05 M sulfuric acid was used as solvent (details of chemicals are given in the manuscript). A solution of 0.05 M sulfuric acid was obtained by diluting concentrated sulfuric acid in an appropriate volume of redistilled water. The required volume of concentrated acid was calculated using the density and percent content indicated by the manufacturer on the original bottle.

b) Quinine sulfate (QS) in 0.05 M sulfuric acid (concentration series Q)
The QS stock solution was prepared by first dissolving an arbitrary amount of QS in 0.05 M sulfuric acid. The resulting solution has a very high absorbance, so aliquots of this solution were added to 0.05 M sulfuric acid to obtain the maximum absorbance A ex ~ 2 in the concentration series. The actual concentration of QS for each point was calculated in triplicate from the absorbance measurements.

c) Quinine sulfate (QS) in 0.05 M sulfuric acid (concentration series Q-f and Q-v).
The procedure was the same as for the Q concentration series, except that the maximum absorbance of the QS was A ex ~ 1 in the concentration series Q-f and Q-v.

d) Potassium dichromate (PD) in 0.05 M sulfuric acid (concentration series Q-f and Q-v)
The PD stock solution was prepared by first dissolving an arbitrary amount of PD in 0.05 M sulfuric acid and diluting it so that the maximum absorbance of the stock solution was A ex ~ 5. For experiments with fixed total concentration of PD, aliquots of this stock solution were added to obtain the constant absorbance A ex ~ 1 for PD in all samples in the Q-f concentration series. For experiments with variable total concentration of PD, different aliquots of this stock solution and 0.05 M sulfuric acid were added to obtain increasing absorbance to the maximum of A ex ~ 1 for PD in the Q-v concentration series.

e) Spectral measurements
UV/Vis spectra were measured using Varian Cary 50 spectrophotometer (Varian, Australia) in a quartz cuvette (l = 1 cm) at room temperature. Values for QS were normalized to the reference value of ε 345 = 5700 M -1 cm -1 in 0.05 M sulfuric acid and are given in Table S4. 1 Table S4.
The fluorescence spectrum of QS was measured at room temperature using Olis RSM 1000F spectrofluorometer (Olis, USA). The excitation wavelength was 345 nm (A 345 ≈ 1) and the excitation bandwidth was 13 nm. The fluorescence units (f.u.) correspond to the ratio of signals obtained from sample and reference PMTs. The fluorescence spectrum was normalized to the maximum value obtained at 452 nm.     .
Eq S1 Values closer to 1 indicate a better fit.

b) Standard error of the estimate, s y
This represents the measure of variation used to check the accuracy of the predictions made with the regression line, and is defined as: 4 where n is the number of data points for linear interpolation. Values closer to 0 indicate a better fit.

c) Limit of detection, LOD
This is defined as the least amount of a substance that can be distinguished from the blank (i.e. absence of the substance) at a given confidence level, i.e. probability of false positive error (α) or false negative error (β).
The background-corrected signal, y SAMPLE -y BLANK , is proportional to the sample concentration c: where y BLANK is the signal from the blank sample and m is the slope of the calibration line.
Limit of detection is then defined as: 5 where s y is the standard error of the estimate (eq. S2) and n is chosen depending on the confidence level required.
For a chosen confidence level of 5 % (i.e., α = β = 0.05), the eq. S5 amounts to: 3 Eq. S6 Considering that the ideal fluorescence signal, IFS, which corresponds to the linear relationship between F and A in the absence of IFE, is a line with slope a = 1 and intercept b = 0 for normalized data, the eq. S3 simplifies to: 3,8,9 where a is the slope of the linear regression line for normalized data ( Table 1 in the manuscript). Values closer to 0 indicate a better fit.

Error estimation a) Error estimation for absorbance (A), uncorrected fluorescence intensity (F 1 and F 2 ) and light path length (h)
The sample standard deviations, s, were estimated for all absorbance and fluorescence intensity measurements (denoted as x i ) as shown in equation: Eq. S8 The sample variances were calculated using the equation: .

Eq. S9
Background corrections were performed for all absorbance and fluorescence intensity measurements. Values measured at zero concentrations of the fluorophore were subtracted from each data point in the fluorophore concentration series. The variance of each background-corrected data point (x BC ) was calculated as the sum of the data point (x) variance and the background (x B ) variance: var( BC ) = var( ) + var( B ).
Eq. S10 We have assumed that there is no correlation between the background and the errors of the titration data points. The main source of variability is likely due to the pipetting errors, which are expected to be random and uncorrelated.
Eq. S11 For aqueous solutions, the path length can be calculated from the absorbance values for water in the nearinfrared wavelength range (900 nm to 1000 nm) using a cuvette and the corresponding microplate. The estimation of h was made by measuring the path length of pure water with parameters: test wavelength λ = 977 nm, reference wavelength λ = 900 nm and correction factor value of 0.186. The correction factor is defined as the absorbance value of water at the test wavelength corrected by the absorbance value of water at the reference wavelength for a path length of 1 cm. 10 The path length of pure water was measured in decaplicate and the average value of the path length obtained for a sample volume of 200 µL was used as the h-estimator.

b) Error estimation for the exponential term (N)
The exponential coefficient in eqs. 5 and 6 in the manuscript can be written as: Eq. S12 The uncertainties in the geometric parameters f and z were not considered in the error estimation calculations. We did not have numerical values for these uncertainties, which result from the tolerances in the manufacture of microplates. The Tecan Spark M10 multimode microplate reader software displays the parameter z in 5 significant figures. The parameter f is a spatial dimension that should be easily measured with high precision.
It is reasonable to assume that the variations of these parameters are statistically insignificant compared to the variation of the parameter h caused by pipetting errors. Therefore, the standard deviation of the term N is calculated to be equal to the standard deviation of the parameter h, assuming that the standard deviations of the parameters f, z 1 , and z 2 are statistically insignificant: Eq. S13 The values of s(N) were calculated only for combinations of z-positions that gave the best results in the correction procedure and were used for comparison with uncorrected (F 1 ) and absorbance-corrected fluorescence (F A ).

c) Error propagation for the absorbance IFE correction (F A )
The correction function given in eq. 1 is a function of three variables and can be written as follows: Eq. S14 The partial derivatives of the above function were calculated with respect to all three variables as: Eq. S15 Eq. S16 Eq. S17 The error propagation was estimated using the following expressions: 4 Eq. S18 Eq. S19 The first expression (eq. S18) is used to calculate the standard deviation for the F A values without considering the covariance factors. The second expression (eq. S19) considers the covariance terms calculated for all the pairs of variables (F 1 , A em , A ex ). The covariance terms seem to be significant in the total sum, considering that it is reasonable to assume that errors in the three variables are not independent of each other, since they are probably the result of pipetting errors and/or geometric imperfections and/or contamination of the microplate. Therefore, eq. S19 was used to calculate the standard deviation for the F A values.

S14 d) Error propagation for the z-position IFE correction (ZINFE, F Z )
The correction function given in eqs. 5 and S12 is a function of three variables, and can be written as follows: The exponential term N is a constant for each IFE correction procedure, since it is a function of the pairs of z-position values (z 1 and z 2 ) used for the particular correction. The term N varies from one correction to another as the combinations of z-position pairs also vary in this respect.
The partial derivatives of the above function were calculated with respect to all three variables as: The error propagation was estimated using the following expressions: Eq. S24 Similar to eqs. S18 and S19, the eq. S24 was used to calculate the standard deviation for the F Z values without considering the covariance factors. The eq. S25 considers the covariance terms calculated for the F 1 and F 2 values (N is constant for each correction). The covariance term may also be significant in the overall sum, since it depends on the combination of z-positions used for a particular correction. Therefore, eq. S25 was used to calculate the standard deviation for the F Z values.
In general, it can be observed that for the closest pairs of z-positions (in terms of numerical values) there is often a very significant correlation. This can be easily verified by plotting the F 1 vs. F 2 values ( Figure  S7). This shows a very good positive correlation between the measured F values over the entire concentration range for the combination of adjacent z-position measurements (z = 14.6 mm and z = 15 mm, red symbols). However, for the pair of the most distant z-positions (z = 14.6 mm and z = 21 mm, blue symbols), there is significantly worse correlation measured F values, especially at the highest concentrations of the fluorophore.

Experimental data
All averaged triplicate data preformatted for automatic online processing and the results obtained have been permanently archived. 11 In total, there are 6 different datasets for 3 different concentration series (Q, Q-v and Q-f) in 2 different types of microplates (T and NT). For each concentration series, a total of 9 fluorescence measurements were performed using the selected available z-positions (n = 9, Table S1).

Fluorescence data
The measured data and the results of the ZINFE correction (F Z ) and the NINFE correction (F N ) are summarized in Figures S8, S9 and S10.
All plots in these figures were created using the JavaScript open-source graphing library Plotly 12 in the online calculator available at https://ninfe.science. 13 Due to incompatible algorithms, two separate online calculators were created: (i) for the proposed ZINFE and NINFE correction, and (ii) for the absorbance IFE correction. The online service requires the properly formatted fluorescence measurements and z-position data (both for NINFE and ZINFE), as well as known geometric parameters for the specific microplate and microplate reader (for ZINFE only).   ; Q-f corresponds to the fixed total concentration of PD (variable ratio of the total concentrations of PD and QS), see manuscript for details. 2 T corresponds to UV-transparent microplates; NT corresponds to non-transparent microplates. 3 F 1 corresponds to uncorrected fluorescence; F Z corresponds to ZINFE-corrected fluorescence intensity (eq. 5); F A corresponds to absorbance IFE-corrected fluorescence intensity (eq. 1); F N corresponds to NINFE-corrected fluorescence intensity. 4 Standard error of the estimate defined by eq. S2. 5 Limit of detection (α = β = 0.05); see manuscript for details. 6 LOD values normalized as percentage of c max , see manuscript for details.      Table S12 and manuscript for details. 4 Background-corrected data, copied from Table 1 in the manuscript for clarity. 5 No background correction, i.e. only raw sample fluorescence data was used for IFE correction.    Table S12 and manuscript for details. Table S22. Overview of the total change of absorbance, ΔA, for all concentration series, calculated from the data shown in Figure S11.

Conflicts of interest
There are no conflicts of interest to declare.